TL;DR
You can't retrieve a private key from a GitHub account, but you can if you explicitly add the key file to a published repository. GitHub also limits you to seeing the fingerprint of public keys attached to an account; to my knowledge, you can't view an account's public keys on GitHub, either.
Private vs. Public Keys
An SSH public key is stored on a single line, and starts with a key-type field such as ssh-rsa or ecdsa-sha2-nistp256. A private key has multiple lines, and a header that identifies the key block. For example:
-----BEGIN RSA PRIVATE KEY-----
GitHub Only Accepts Public Keys
You can't actually add a valid private key to a GitHub account anyway. Currently, GitHub only accepts RSA and DSA public keys for this purpose. If you try to add a different key-type, or an invalid public key of any sort, you receive an error like the following:
Key is invalid. It must begin with 'ssh-rsa' or 'ssh-dss'. Check that you're copying the public half of the key
Private Keys Are Not Stored Server-Side in GitHub Accounts
The upshot of all this is that--quite aside from the fact that you should never store your private keys server-side anyway--it isn't possible to attach a private key to a GitHub account. Therefore, there's no way to retrieve the private half of a key-pair from a GitHub account.
Of course, while you can't add SSH private keys to a GitHub account, you can add them to a published repository. As an example of this, the Vagrant project publishes a full key-pair, but it's a special case.
Publishing your secret key is inherently insecure. Don't do it unless you understand the implications.